Nguyen Ngoc Cuong, Hölder Continuous Solutions to Complex Monge-Ampère Equations and its Applications II

The Monge-Ampère equations provide Kähler-Einstein metrics on projective manifolds with negative or zero first Chern classes thanks to the AubinYau and Yau theorems. However, most projective manifolds do not have a negative definite or trivial first Chern class. The study of the canonical metric on these manifolds leads to study degenerate Monge-Ampère equations both on the right hand side and on the background form. It turns out that Hölder continuity is the best regularity we can hope for the solution (quasi-plurisubharmonic potential) to the equation. We discuss situations and criterions such that we will have this property and some applications.

More precisely, let X be a compact Kähler manifold of dimension n and ω a Kähler form on X. We consider the complex Monge-Ampère equation (dd^cu+ω)ⁿ = µ, where µ is a given positive measure on X of suitable mass and u is an ω-plurisubharmonic function. We show that the equation admits a Hölder continuous solution if and only if the measure µ, seen as a functional on a complex Sobolev space W^∗(X), is Hölder continuous. Here, denote by W^1,2(X) the Sobolev space of real valued functions f on X such that both f and df are of class L²(X). Then, the complex Sobolev space W^∗(X), introduced by Dinh-Sibony, is the space of all functions f ∈ W^1,2(X) such that

df ∧ d^cf ≤ T

for some closed positive (1, 1)-current T on X.

A similar result is also obtained for the complex Monge-Ampère equations on domains of Cⁿ.

In the first talk we give motivations and background to study weak solutions of Monge-Ampère equations.

In the second talk we focus on the several criterions such that the solution is Hölder continuous, or alternatively we give the estimate of Hölder norm that depends very weak on the datum such as L^p-norm, p > 1, of the right hand sides.

This is based on joint work with Tien-Cuong Dinh and Slawomir Ko lodziej.